2007 AMC 10B Problem 5

Question

In a certain land, all Arogs are Brafs, all Crups are Brafs, all Dramps are Arogs, and all Crups are Dramps. Which of the following statements is implied by these facts?

All Dramps are Brafs and are Crups.

All Brafs are Crups and are Dramps.

All Arogs are Crups and are Dramps.

All Crups are Arogs and are Brafs.

All Arogs are Dramps and some Arogs may not be Crups.

Answer: D

Concepts:logical deduction

Difficulty rating: 1020

Small hint:

Write each fact as an implication between the categories

Big hint:

Follow the chain

C\Rightarrow D\Rightarrow A\Rightarrow B

Solution:

Writing the statements as implications, being a Crup implies being a Dramp, a Dramp implies being an Arog, and an Arog implies being a Braf:

C\Rightarrow D\Rightarrow A\Rightarrow B.

So every Crup is a Dramp, an Arog, and a Braf. The only listed statement guaranteed true is that all Crups are Arogs and Brafs.

Thus, the correct answer is D.